Most people, when they think of Nicolas Bourbaki, think first of his multi-volume "textbook", the Éléments de Mathématique. It may well be the case, however, that Bourbaki's most significant contribution to mathematics has been the long-running Séminaire that bears his name. Held each academic year in November, March, and June, the seminar is a venue for high-level exposition. Its yearly collection of "exposés" is a reliable source of information about what is going on in mathematics, at least from the point of view of the Bourbakistes.
The structure of the seminar is now standard: top mathematicians are invited to give talks, usually not about their own work, but about someone else's recent breakthrough. Important new theorems or concepts serve as the topics, so that the volumes of the seminar notes also serve as a sketch of the recent history of mathematics as it appeared each year.
The Séminaire has a complex publishing history, but for the last twenty or so years it has been published in the Société Mathématique de France's Asterisque series. The volume under review contains the notes from talks given in the 2005–06 academic year. The topics covered are quite varied, but show a slant towards traditional interests of Bourbaki: algebraic geometry and number theory are well represented, as are Lie theory and differential equations. But the perfect graph conjecture (here called the perfect graph theorem) gets an exposé, showing that the organizers are not restricted to those areas. The exposition is at a very high level, quite technical but often illuminating. Most of the articles are in French, of course.
This is an important series that most libraries should have.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.
952 Michel Brion, Compactification de l'espace de modules des variétés abéliennes principalement polarisées [d'après V. Alexeev]
953 Nicolas Burq, Explosion pour l'équation de Schrödinger au régime du "log log" [d'après Merle-Raphael]
954 Daniel Huybrechts, Projectivity of Kähler manifords — Kodaira's problem [after C. Voisin]
955 Christophe Soulé, Genres de Todd et valeurs aux entiers des dérivées de fonctions L
956 Jean-Pierre Wintenberger, La conjecture de modularité de Serre: las cas de conducteur 1 [d'après C. Khare]
957 Gérard Cornuéjols, Le théoreme fort des graphes parfaits
958 Antoine Ducros, Espaces analytiques p-adiques au sens de Berkovich
959 Emmanuel Kowalski, Écarts entre nombres premiers successifs [d'après Goldston, Pintz, Yildirim,…]
960 Nicolas Lerner, The verification of Nirenberg-Treves conjecture
961 Stefaan Vaes, Rigidity results for Bernoulli actions and their von Neumann algebras [after Sorin Popa]
962 Michael Christ, Modulation invariant and multilinear singular integral operators [after Lacey and Thiele]
963 Caroline Gruson, Sure les représentations de dimension finie de la super-algèbre de Lie gl(m,n) [d'après Serganova]
964 William Messing, Travaux de Zink
965 Igor Rodnianski, The Wave Map Problem. Small data critical regularity [after T. Tao]
966 Jean-Christophe Yoccoz, Ensembles de Julia de mesure positive et disques de Siegel des polynomes quadratiques [d'après X. Buff et A Chéritat]